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By giving the longitudes of points in the equatorial region on both charts. In a very similar fashion, one can give a fully nonsingular representation of the gauge potential over the whole surface surrounding a monopole provided that one pays the price of using more than one coordinate patch and specifying in the overlap regions of these patches the relation between the potentials defined in the different patches. Clearly, just as for the cartographer's zenithal projections, such patched representations of gauge potentials are not unique; apart from convenience, it is immaterial whether one chooses to cover the surface of the sphere by two or more patches, or what size and shape one assigns to each patch.
Theories with gauge group SU(N) have no monopoles, but the pure gauge theories with gauge group SU(N)/%N will have monopoles whose charges take values in jy. e. integers modulo 3, or equivalently the cube roots of unity: = expi27rr/3, r = 0,1,2. As a further example, consider the standard electroweak theory. 9). 2. NONABELIAN MONOPOLES 27 SU(2) and the horizontal axis the group (7(1), and the parallel edges of the rectangle are understood to be identified so as to make the rectangle into a 'hyper-torus'.
It is instructive at this point to try to visualize what patching of the monopole potential means. We recall that the phase of the wave function i/> of an (electrically) charged particle can range from 0 to 2x and that it is the gauge potential a ß which specifies what value of this phase at one space-time point should be regarded as being 'parallel' to a given value at a neighbouring point. So we can imagine that to every point x there is attached a circle representing the various values that the phase of if) can take, and a ^ x ) is what specifies which point on the circle at x + dx^ corresponds to which point on the circle at x.